Question

what are the values of a and b​

Answer 1

Answer:

$\large\boxed{B.\ a=8,\ b=4\sqrt5}$

Step-by-step explanation:

We have three similar triangles:

$\triangle ABC\sim\triangle DBA\sim\triangle DAC$

therefore the sides are in proportion:

$\dfrac{AD}{CD}=\dfrac{BD}{AD}$

We have AD = a, CD = 16 and BD = 4. Substitute:

$\dfrac{a}{16}=\dfrac{4}{a}$              cross multiply

$(a)(a)=(16)(4)\\\\a^2=64\to a=\sqrt{64}\\\\a=8$

and other proportion:

$\dfrac{AB}{BC}=\dfrac{BD}{AB}$

We have AB = b, BC = 16 + 4 = 20 and BD = 4. Substitute:

$\dfrac{b}{20}=\dfrac{4}{b}$             cross multiply

$(b)(b)=(20)(4)\\\\b^2=80\to b=\sqrt{80}\\\\b=\sqrt{16\cdot5}\\\\b=\sqrt{16}\cdot\sqrt{5}\\\\b=4\sqrt{5}$

Answer 2

Option B is the correct option.

Similarity of two triangles,

Condition for the similarity of two triangles,

"If two triangles are similar, corresponding sides of these triangles will be proportional"

From the picture attached,

ΔADC and ΔBDA are similar,

By the condition of the similarity, corresponding sides will be proportional.

$\frac{DA}{BD}= \frac{DC}{DA}= \frac{AC}{BA}$

Substitute the measures given in the picture,

$\frac{a}{4}= \frac{16}{a}= \frac{AC}{b}$

$\frac{a}{4}=\frac{16}{a}$

$a^2=64$

$a=8$

Now apply Pythagoras theorem in ΔADB,

AB² = AD² + DB²

b² = a² + 4²

b² = 8² + 4² [Since, a = 8]

b = √80

b = 4√5

    Therefore, Option B will be the correct option.

Learn more about the similarity of two triangles here,

https://brainly.com/question/25187489?referrer=searchResults